R12 Self-Authenticating State No-Go

Status: exact fault-model theorem; reject as a novel reasoning mechanism.

Candidate

Carry a compact causal state together with a locally checkable certificate so each reasoning step detects or repairs corruption before it compounds.

Collapse theorem

Let E:X->{0,1}^N encode causal states and let V:{0,1}^N -> X union {reject} satisfy V(E(x))=x.

1. If every corruption of at most t bits must avoid acceptance as a different state, then distinct codewords have Hamming distance at least t+1.
2. If every such corruption must be corrected to the original state, the distance is at least 2t+1.
3. Against unrestricted substitution, public self-authentication is impossible: replacing E(x) with another valid E(y) passes completeness. Detection therefore needs a bounded-distance fault model or an external root, secret, counter, checkpoint, or trusted prior state.
4. A recurrent control with the same N bits and transition work can execute the identical map z -> E(U_a(D(z))), including verification and recovery.

The first two statements are exactly error-detecting and error-correcting code distance. The third identifies the hidden trust source. The fourth is a resource-preserving identity simulation under the corrected R12 gate.

Smallest witnesses

For one causal bit with update x <- x xor a, the code 0->00, 1->11 is the smallest one-bit-error detector. The repetition code 0->000, 1->111 with majority decoding is the smallest one-bit-error corrector. A recurrent control given two or three bits and the same repair work reproduces either exactly.

Under independent boundary noise BSC(p), threefold repetition fails per step with probability 3p^2-2p^3. Longer codes can extend the reliable horizon, but the resource is redundancy plus a trusted repair boundary.

Prior-art and resource boundary

• accepted packets form an error-detecting/correcting code;
• constant-query local checks are locally testable codes and import proof-oracle storage plus soundness error;
• noisy verification/repair is fault-tolerant computation;
• recursive execution certificates are proof-carrying data or incrementally verifiable computation and certify a specified update, not its semantic truth;
• detection without correction is checkpoint/restart or fail-stop recovery;
• cryptographic authentication imports a key/root and replay protection imports a trusted counter or history commitment.

The only exposed resource is fault-domain-separated trust. Reopen only for a separation against coded/proof-carrying controls with identical bits, precision, trusted boundaries, checkpoints, FLOPs, and correlated-noise exposure. No CPU falsifier, Shohin fit, or H100 job is authorized.